library(tidyverse)
options(dplyr.summarise.inform = FALSE)
For this lab you will use multivariate auto-regressive state-space (MARSS) to analyze multivariate salmon data from the Columbia River. These data are noisy and gappy. They are estimates of total spawner abundance and might include hatchery spawners.
These data are from the Coordinated Assessments
Partnership (CAP) and downloaded using the rCAX R client for the
CAX (the CAP database) API. The data are saved in
Lab-2/Data_Images/columbia-river.rda.
load(here::here("Lab-2", "Data_Images", "columbia-river.rda"))
The data set has data for fi endangered and threatened ESU (Evolutionary Significant Units) in the Lower Columbia River.
esu <- unique(columbia.river$esu_dps)
esu
## [1] "Steelhead (Middle Columbia River DPS)"
## [2] "Steelhead (Upper Columbia River DPS)"
## [3] "Steelhead (Lower Columbia River DPS)"
## [4] "Salmon, coho (Lower Columbia River ESU)"
## [5] "Salmon, Chinook (Lower Columbia River ESU)"
Figure from ESA recovery plan for Lower Columbia River Coho salmon, Lower Columbia River Chinook salmon, Columbia River Chum salmon, and Lower Columbia River steelhead. 2013. NMFS NW Region. https://repository.library.noaa.gov/view/noaa/16002
The dataset has the following columns
colnames(columbia.river)
## [1] "species" "esu_dps" "majorpopgroup" "esapopname"
## [5] "commonpopname" "run" "spawningyear" "value"
## [9] "value_type"
Let’s load one ESU and make a plot. Create a function.
plotesu <- function(esuname){
df <- columbia.river %>% subset(esu_dps %in% esuname)
ggplot(df, aes(x=spawningyear, y=log(value), color=majorpopgroup)) +
geom_point(size=0.2, na.rm = TRUE) +
theme(strip.text.x = element_text(size = 3)) +
theme(axis.text.x = element_text(size = 5, angle = 90)) +
facet_wrap(~esapopname) +
ggtitle(paste0(esuname, collapse="\n"))
}
plotesu(esu[3])
plotesu(esu[4])
plotesu(esu[5])
plotesu(esu[1])
df <- columbia.river %>% subset(species == "Chinook salmon")
ggplot(df, aes(x=spawningyear, y=log(value), color=run)) +
geom_point(size=0.2, na.rm = TRUE) +
theme(strip.text.x = element_text(size = 3)) +
theme(axis.text.x = element_text(size = 5, angle = 90)) +
facet_wrap(~esapopname)
Create estimates of spawner abundance for all missing years and provide estimates of the decline from the historical abundance.
Evaluate support for the major population groups. Are the populations in the groups more correlated than outside the groups?
Evaluate the evidence of cycling in the data. We will talk about how to do this on the Tuesday after lab.
Simplify
If your ESU has many populations, start with a smaller set of 4-7 populations.
Assumptions
You can assume that R="diagonal and equal" and
A="scaling". Assume that “historical” means the earliest
years available for your group.
States
Your abundance estimate is the “x” or “state” estimates. You can get this from
fit$states
or
tsSmooth(fit)
where fit is from fit <- MARSS()
plotting
Estimate of the mean of the spawner counts based on your x model.
autoplot(fit, plot.type="fitted.ytT")
diagnostics
autoplot(fit, plot.type="residuals")
Describe your assumptions about the x and how the data time series are related to x.
Write out your assumptions as different models in matrix form, fit each and then compare these with AIC or AICc.
Do your estimates differ depending on the assumptions you make about the structure of the data, i.e. you assumptions about the x’s, Q, and U.
Here I show how I might analyze the Upper Columbia Steelhead data.
Figure from 2022 5-Year Review: Summary & Evaluation of Upper Columbia River Spring-run Chinook Salmon and Upper Columbia River Steelhead. NMFS. West Coast Region. https://doi.org/10.25923/p4w5-dp31
Set up the data. We need the time series in a matrix with time across the columns.
Load the data.
load(here::here("Lab-2", "Data_Images", "columbia-river.rda"))
Wrangle the data.
library(dplyr)
esuname <- esu[2]
dat <- columbia.river %>%
subset(esu_dps == esuname) %>% # get only this ESU
mutate(log.spawner = log(value)) %>% # create a column called log.spawner
select(esapopname, spawningyear, log.spawner) %>% # get just the columns that I need
pivot_wider(names_from = "esapopname", values_from = "log.spawner") %>%
column_to_rownames(var = "spawningyear") %>% # make the years rownames
as.matrix() %>% # turn into a matrix with year down the rows
t() # make time across the columns
# MARSS complains if I don't do this
dat[is.na(dat)] <- NA
Clean up the row names
tmp <- rownames(dat)
tmp <- stringr::str_replace(tmp, "Steelhead [(]Upper Columbia River DPS[)]", "")
tmp <- stringr::str_replace(tmp, "River - summer", "")
tmp <- stringr::str_trim(tmp)
rownames(dat) <- tmp
Specify a model
mod.list1 <- list(
U = "unequal",
R = "diagonal and equal",
Q = "unconstrained"
)
Fit the model. In this case, a BFGS algorithm is faster.
library(MARSS)
fit1 <- MARSS(dat, model=mod.list1, method="BFGS")
## Success! Converged in 235 iterations.
## Function MARSSkfas used for likelihood calculation.
##
## MARSS fit is
## Estimation method: BFGS
## Estimation converged in 235 iterations.
## Log-likelihood: -109.4078
## AIC: 256.8155 AICc: 262.1676
##
## Estimate
## R.diag 0.00997
## U.X.Entiat 0.02182
## U.X.Methow 0.01852
## U.X.Okanogan 0.00140
## U.X.Wenatchee -0.02222
## Q.(1,1) 0.28016
## Q.(2,1) 0.12303
## Q.(3,1) 0.14275
## Q.(4,1) 0.23415
## Q.(2,2) 0.31642
## Q.(3,2) 0.30806
## Q.(4,2) 0.19061
## Q.(3,3) 0.31031
## Q.(4,3) 0.18852
## Q.(4,4) 0.52813
## x0.X.Entiat 4.61647
## x0.X.Methow 6.43401
## x0.X.Okanogan 6.47217
## x0.X.Wenatchee 8.04868
## Initial states (x0) defined at t=0
##
## Standard errors have not been calculated.
## Use MARSSparamCIs to compute CIs and bias estimates.
Hmmmmm, the Q variance is so high that it perfectly fits the data. That doesn’t seem right.
autoplot(fit1, plot.type="fitted.ytT")
## MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.
## plot.type = fitted.ytT
## Finished plots.
Let’s look at the corrplot. Interesting. The Methow and Entiat are almost perfectly correlated while the Entiat and Wenatchee are somewhat correlated. That makes sense if you look at a map.
library(corrplot)
## corrplot 0.92 loaded
Q <- coef(fit1, type="matrix")$Q
corrmat <- diag(1/sqrt(diag(Q))) %*% Q %*% diag(1/sqrt(diag(Q)))
corrplot(corrmat)
I need to use the EM algorithm (remove method="BFGS")
because the BFGS algorithm doesn’t allow constraints on the Q
matrix.
mod.list2 <- list(
U = "unequal",
R = "diagonal and equal",
Q = "equalvarcov"
)
fit2 <- MARSS(dat, model=mod.list2, control = list(maxit=1000))
## Success! abstol and log-log tests passed at 794 iterations.
## Alert: conv.test.slope.tol is 0.5.
## Test with smaller values (<0.1) to ensure convergence.
##
## MARSS fit is
## Estimation method: kem
## Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
## Estimation converged in 794 iterations.
## Log-likelihood: -120.6028
## AIC: 263.2057 AICc: 264.9657
##
## Estimate
## R.diag 0.1290
## U.X.Entiat 0.0257
## U.X.Methow 0.0311
## U.X.Okanogan 0.0166
## U.X.Wenatchee -0.0282
## Q.diag 0.2632
## Q.offdiag 0.2631
## x0.X.Entiat 4.2026
## x0.X.Methow 5.9042
## x0.X.Okanogan 5.8359
## x0.X.Wenatchee 8.0703
## Initial states (x0) defined at t=0
##
## Standard errors have not been calculated.
## Use MARSSparamCIs to compute CIs and bias estimates.
autoplot(fit2, plot.type="fitted.ytT")
## MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.
## plot.type = fitted.ytT
## Finished plots.
Now I want try something different. I will treat the Methow-Okanogan as one state and the Entiat-Wenatchee as another. I’ll let these be correlated together. Interesting, these two are estimated to be perfectly correlated.
mod.list3 <- mod.list1
mod.list3$Q <- "unconstrained"
mod.list3$Z <- factor(c("ew", "mo", "mo", "ew"))
fit3 <- MARSS(dat, model = mod.list3)
## Warning! Reached maxit before parameters converged. Maxit was 500.
## neither abstol nor log-log convergence tests were passed.
##
## MARSS fit is
## Estimation method: kem
## Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
## WARNING: maxit reached at 500 iter before convergence.
## Neither abstol nor log-log convergence test were passed.
## The likelihood and params are not at the ML values.
## Try setting control$maxit higher.
## Log-likelihood: -137.532
## AIC: 295.064 AICc: 296.5209
##
## Estimate
## A.Okanogan -0.68779
## A.Wenatchee 1.54127
## R.diag 0.18062
## U.ew -0.02175
## U.mo 0.00374
## Q.(1,1) 0.22050
## Q.(2,1) 0.22103
## Q.(2,2) 0.22164
## x0.ew 6.51468
## x0.mo 7.33795
## Initial states (x0) defined at t=0
##
## Standard errors have not been calculated.
## Use MARSSparamCIs to compute CIs and bias estimates.
##
## Convergence warnings
## Warning: the logLik parameter value has not converged.
## Type MARSSinfo("convergence") for more info on this warning.
autoplot(fit3, plot.type="fitted.ytT")
## plot.type = fitted.ytT
## Finished plots.
Finally, let’s look at the AIC values. Fit1 was very flexible and can put a line through the data so I know I have at least one model in the set that can fit the data. Well, the most flexible model is the best. At this point, I’d like to look just at data after 1980 or so. I don’t like the big dip that happened in the Wenatchee River. I’d want to talk to the biologists to find out what happened, especially because I know that there might be hatchery releases in this system.
aic <- c(fit1$AICc, fit2$AICc, fit3$AICc)
aic-min(aic)
## [1] 0.00000 2.79807 34.35331
Let’s just look at the data after 1987 to eliminate that string of NAs in the 3 rivers.
dat87 <- dat[,colnames(dat)>1987]
Let’s look the acf to look for evidence of cycling. Due to the nature of their life-cycle where they tend to return back to their spawning grounds after a certain numbers of years, we might expect some cycling although steelhead aren’t really known for this (unlike sockeye, chinook and pink).
Well no obvious cycles.
par(mfrow=c(2,2))
for(i in 1:4){
acf(dat87[i,], na.action=na.pass, main=rownames(dat87)[i])
}
But let’s go through how we might include cycles. We are going to include cycles with frequency 3, 4, and 5, choosem to reflect steelhead returning after 3, 4 or 5 years.
TT <- dim(dat87)[2] #number of time steps
covariates <- rbind(
forecast::fourier(ts(1:TT, freq=3), K=1) |> t(),
forecast::fourier(ts(1:TT, freq=4), K=1) |> t(),
forecast::fourier(ts(1:TT, freq=5), K=1) |> t()
)
## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
Now let’s fit a model with these covariates. Let’s analyze the populations separately, so Q is diagonal.
mod.list4 <- list(
Q = "unconstrained",
U = "unequal",
R = "diagonal and equal",
D = "unconstrained",
d = covariates
)
fit4.87 <- MARSS(dat87, model=mod.list4)
## Success! abstol and log-log tests passed at 78 iterations.
## Alert: conv.test.slope.tol is 0.5.
## Test with smaller values (<0.1) to ensure convergence.
##
## MARSS fit is
## Estimation method: kem
## Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
## Estimation converged in 78 iterations.
## Log-likelihood: -55.48472
## AIC: 196.9694 AICc: 238.5519
##
## Estimate
## R.diag 0.00841
## U.X.Entiat -0.01592
## U.X.Methow 0.00629
## U.X.Okanogan -0.01331
## U.X.Wenatchee -0.06327
## Q.(1,1) 0.21426
## Q.(2,1) 0.10446
## Q.(3,1) 0.12493
## Q.(4,1) 0.12760
## Q.(2,2) 0.21888
## Q.(3,2) 0.21364
## Q.(4,2) 0.13562
## Q.(3,3) 0.22037
## Q.(4,3) 0.13483
## Q.(4,4) 0.31566
## x0.X.Entiat 6.34777
## x0.X.Methow 7.39581
## x0.X.Okanogan 7.02470
## x0.X.Wenatchee 8.65239
## D.(Entiat,S1-3) -0.03464
## D.(Methow,S1-3) -0.12969
## D.(Okanogan,S1-3) -0.11592
## D.(Wenatchee,S1-3) -0.01482
## D.(Entiat,C1-3) 0.02784
## D.(Methow,C1-3) -0.08604
## D.(Okanogan,C1-3) -0.09585
## D.(Wenatchee,C1-3) 0.05808
## D.(Entiat,S1-4) -0.11286
## D.(Methow,S1-4) -0.13983
## D.(Okanogan,S1-4) -0.09480
## D.(Wenatchee,S1-4) -0.06365
## D.(Entiat,C1-4) 0.02030
## D.(Methow,C1-4) -0.09692
## D.(Okanogan,C1-4) -0.08208
## D.(Wenatchee,C1-4) -0.08237
## D.(Entiat,S1-5) -0.19272
## D.(Methow,S1-5) 0.05745
## D.(Okanogan,S1-5) 0.07723
## D.(Wenatchee,S1-5) -0.18255
## D.(Entiat,C1-5) -0.01818
## D.(Methow,C1-5) 0.17916
## D.(Okanogan,C1-5) 0.15510
## D.(Wenatchee,C1-5) -0.02965
## Initial states (x0) defined at t=0
##
## Standard errors have not been calculated.
## Use MARSSparamCIs to compute CIs and bias estimates.
Let’s plot the estimates. broom::tidy() will get a data
frame with the terms, estimates and CIs.
library(broom)
df <- tidy(fit4.87) %>%
subset(stringr::str_sub(term,1,1)=="D")
df$lag <- as.factor(rep(3:5, each=8))
df$river <- as.factor(rep(rownames(dat87),3))
df$sc <- rep(rep(c("S","C"), each=4), 3)
df$type <- paste0(df$sc,df$lag)
We can then plot this. Interesting. Some support for 5 year cycles.
ggplot(df, aes(x=type, y=estimate, col=lag)) +
geom_point() +
geom_errorbar(aes(ymin=conf.low, ymax=conf.up), width=.2, position=position_dodge(.9)) +
geom_hline(yintercept = 0) +
facet_wrap(~river) +
ggtitle("The cycle estimates with CIs")
Let’s compare some other models.
# No cycles
mod.list <- list(
Q = "unconstrained",
U = "unequal",
R = "diagonal and equal"
)
fit1.87 <- MARSS(dat87, model=mod.list, silent=TRUE)
# Only lag 5 cycles
mod.list <- list(
Q = "unconstrained",
U = "unequal",
R = "diagonal and equal",
D = "unconstrained",
d = covariates[5:6,]
)
fit5.87 <- MARSS(dat87, model=mod.list, silent=TRUE)
# Cycles in the process
# which doesn't really make sense for salmon since the cycles are age-structure cycles
# which act like cycles in the observations
mod.list <- list(
Q = "unconstrained",
U = "unequal",
R = "diagonal and equal",
C = "unconstrained",
c = covariates
)
fit6.87 <- MARSS(dat87, model=mod.list, silent=TRUE)
Hmm model without cyles is much better (lower AICc). Even if we only
have the 5 year cycles (covariates[5:6,]), the AICc is
larger than for the models with cycles.
aic <- c(fit1.87$AICc, fit4.87$AICc, fit5.87$AICc, fit6.87$AICc)
aic-min(aic)
## [1] 0.00000 56.99612 11.66091 56.99461
Chapter 7 MARSS models. ATSA Lab Book. https://atsa-es.github.io/atsa-labs/chap-mss.html
Chapter 8 MARSS models with covariate. ATSA Lab Book. https://atsa-es.github.io/atsa-labs/chap-msscov.html
Chapter 16 Modeling cyclic sockeye https://atsa-es.github.io/atsa-labs/chap-cyclic-sockeye.html